On linear preservers of semipositive matrices

Given proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, an $m \times n$ matrix $A$ with real entries is said to be semipositive if there exists a $x \in K_1^{\circ}$ such that $Ax \in K_2^{\circ}$, where $K^{\circ}$ denotes the interior of a proper cone $K$. This set is denoted by $S(K_1,K_2)$. We resolve a recent conjecture on the structure of into linear preservers of $S(\mathbb{R}^n_+,\mathbb{R}^m_+)$. We also determine linear preservers of the set $S(K_1,K_2)$ for arbitrary proper cones $K_1$ and $K_2$. Preservers of the subclass of those elements of $S(K_1,K_2)$ with a $(K_2,K_1)$-nonnegative left inverse as well as connections between strong linear preservers of $S(K_1,K_2)$ with other linear preserver problems are considered.

Linear k-power Preservers on Tensor Products of Matrices

Invariants and the study of the map preserving a certain invariant play vital roles in the study of the theoretical mathematics. The preserver problems are the researches on linear operators that preserve certain invariants between matrix sets. Based on the result of linear $k$-power preservers on general matrix spaces, in terms of the advantages of matrix tensor products which is not limited by the size of matrices as well as the immense actual background, the study of the structure of the linear $k$-power preservers on tensor products of matrices is essential, which is coped with in this paper. That is to characterize a linear map $f:M_{m_{1}\cdots m_{l}}\rightarrow M_{m_{1}\cdots m_{l}}$ satisfying $f(X_{1}\otimes \cdots \otimes X_{l})^{k}=f\left( (X_{1}\otimes \cdots \otimes X_{l})^{k}\right) $ for all $X_{1}\otimes \cdots \otimes X_{l}\in M_{m_{1}\cdots m_{l}}$.

Two linear preserver problems on graphs

Let n, t, k be integers such that 3 ≤ t,k ≤ n. Denote by G_n the set of graphs with vertex set {1,2,...,n}. In this paper, the complete linear transformations on G_n mapping K_t-free graphs to K_t-free graphs are characterized. The complete linear transformations on G_n mapping C_k-free graphs to C_k-free graphs are also characterized when n ≥ 6.